Find the intervals in which the following functions are increasin… - Vidyadip

Question

Find the intervals in which the following functions are increasing or decreasing.
$f(x) = 2x^3 - 24x + 107$

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Answer

$f(x) = 2x^3 - 24x + 107$
$f'(x) = 6x^2 - 24$
$= 6(x^2 - 4)$
$= 6(x + 2)(x - 2)$
For f(x) to be increasing, we must have
$f'(x) > 0$
$\Rightarrow 6(x + 2)(x - 2) > 0$
$\Rightarrow (x + 2)(x - 2) > 0$
[Since, $6 > 0, 6(x + 2)(x - 2) > 0$
$\Rightarrow (x + 2)(x - 2) > 0]$
$\Rightarrow x < -2 or x > 2$
$\Rightarrow\text{x}\in(-\infty,-2)\cup(2,\infty)$
So, f(x) is increasing on $\text{x}\in(-\infty,-2)\cup(2,\infty).$
For f(x) to be decreasing, we must have,
$f'(x) < 0$
$\Rightarrow 6(x + 2)(x - 2) < 0$
$\Rightarrow (x + 2)(x - 2) < 0$
[Since, $6 > 0, 6(x + 2)(x - 2) < 0$
$\Rightarrow (x + 2)(x - 2) < 0]$
$\Rightarrow -2 < x < 2$
$\Rightarrow\text{x}\in(-2,2)$
So, f(x) is decreasing on $\text{x}\in(-2,2).$

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